ADVANCED UNDERGRADUATE LABORATORY
HALL
Semiconductor Resistance,
Band Gap, and Hall Effect
Revisions:
November 2011, January 2016: David Bailey
October 2010: Henry van Driel
February 2006: Jason Harlow
November 1996: David Bailey
March 1990: John Pitre & Taek-Soon Yoon
Copyright © 2011-2016 University of Toronto
This work is licensed under the Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 Unported License.
(http://creativecommons.org/licenses/by-nc-sa/3.0/)
Introduction
Solid materials are usually classified as metals, semiconductors, or insulators. In general, metals
conduct electricity better than semiconductors, and insulators essentially do not conduct electricity at all,
but the critical distinguishing characteristic between the 3 classes of materials is their electronic energy
structure. The basic parameters determining the conductivity of a semiconductor are the band gap, the
charge carrier densities (intrinsic and extrinsic), and the mobility of the charge carriers. The goal of this
experiment is to measure these parameters for two doped germanium samples.
The allowed energies of electrons inside solid materials are quantum mechanically restricted to
certain ranges known as energy bands. For semiconductors the lower energy bands are completely filled
in intrinsic semiconductors at low temperatures and are known as valence bands; the first band that is
normally empty at low temperatures is known as the conduction band. The difference in energy
between the top of the valence band and the bottom of the conduction band is known as the fundamental
energy gap. With an increase in temperature electrons can be thermally activated across the relatively
small band gap of a semiconductor allowing the free carriers to move. Alternatively, doping can add free
carriers to the conduction or valence band. Materials in which an energy band is always partially filled
are metals, with the partially filled bands allows electrons to move freely. Insulators are materials where
the band gap is so large that the conduction band is essentially always empty. As a result, the resistivity
of an insulator is very high.
The resistance of a semiconductor is normally several orders of magnitude higher than that of a
metal, but it is the temperature dependence of the resistance that distinguishes a metal from a
semiconductor. The resistance of the semiconductor germanium (Ge) near room temperature decreases
with increasing temperature, whereas in a metal the resistance generally increases with temperature. In
a metal the number of carriers is fixed, and the increase in resistance of the metal is because of the
increased number of thermally excited lattice vibrations (phonons) from which the charge carriers can
scatter. In a semiconductor the increase in scattering is usually overwhelmed by the exponential
increase in the number of carriers, as a result of thermal excitation across the energy gap. The
temperature dependence of the resistance can be used to determine the band gap.
The Hall voltage is the voltage transverse to both magnetic field and current. It appears when a
magnetic field transverse to the direction of current flow is applied. The sign of the Hall voltage
determines whether the dominant carriers in the semiconductor are electrons or holes; its magnitude is a
measure of the carrier concentration.
Theory
Semiconductors, metals, electrolytes and other conducting materials have charge carriers that are
free to move about in the substance, not being tightly bound to any particular atom or molecule. If an
electric field exists in such a conducting material, positive and/or negative charge carriers tend to drift
towards the electrode or contact of opposite polarity, thus causing an electric current. The charge
carriers are not able to travel unimpeded from one contact to another; in a solid they collide with defects
in the lattice structure such as impurity atoms and lattice vibrations, and in a liquid or gas with other
ions, molecules, or atoms. These collisions impede the current flow and give rise to electrical resistance.
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Conductivity
To explain the conduction processes in a semiconductor we need to assume that the electric
current may be caused by the flow of two types of charged carriers, electrons (in conduction band) and
holes (absence of an electron from valence band). The hole may be looked upon as a particle carrying a
positive electronic charge and its movement is analogous to that of the gap in a line of cars as they
accelerate away from a traffic light which has just turned green: as the cars (electrons) move forward in
turn, the gap (hole) moves in the opposite direction. The electrical conductivity may be expressed in
terms of the number of charge carriers, their charge, and their mobility; the average velocity of the
carrier per unit of electric field is
s = pemh - n(-e)me = pemh + neme
(1)
where p is the number density of holes in m-3, n is the number density of electrons in m-3, e=1.602×10-19
C is the magnitude of the charge of the electron, h is the hole mobility in m2s-1V-1, and e is the electron
mobility in m2s-1V-1.
Equation (1) applies to a semiconductor when the numbers of electrons and holes present are
both significant. If the semiconductor is “doped” by introducing suitable impurities, the charge carriers
can be made predominantly of one type. In germanium, which is in Group 4 of the periodic table, a
Group 5 impurity such as antimony has an extra electron which becomes a free charge carrier and the
semiconductor is then said to be n-type. Conversely, a Group 3 impurity such as indium makes
germanium p-type, since the missing electron is equivalent to a hole in the valence band. Equation (1)
thus becomes
s = pemh
(2)
for a p-type semiconductor or
s = neme
(3)
for an n-type semiconductor.
In a pure semiconductor the number density n or p of charge carriers per unit volume is far
smaller than in a metal. For example, in pure Ge at room temperature, n ≈ p ≈ 1018 m-3, whereas in a
typical metal, n ≈ 1028 m-3. This results in a much larger conductivity in a metal than in a semiconductor.
However as the temperature is raised in a semiconductor the conductivity increases rapidly, whereas in
metal the conductivity decreases slowly. This striking difference in behaviour is a direct consequence of
the property of a semiconductor which distinguishes it from a metal, namely, the existence of a
relatively small energy gap between electron states in the valence band (which correspond to the bound
electrons constituting the covalent bonds between the atoms) and electron states in the conduction band
(which correspond to electrons free to conduct electricity). At low temperatures, all the electrons are in
the valence band which is completely full (corresponding to the saturation of the 4 covalent bonds by the
8 electrons available from each Ge atom of valence 4 and its 4 nearest neighbours), so that there are no
free charges or carriers, except those induced by impurities. Such a semiconductor is then referred to as
extrinsic. As the temperature, T, is raised, the semiconductor becomes “intrinsic”, meaning that the
number density of charge carriers, as determined by the band populations of the atoms, determines the
resistivity.
In an intrinsic semiconductor the number of electrons is equal to the number of holes, because the
thermal excitation of an electron leaves behind a hole in the valence band. It can be shown that:
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(4)
where the subscript i denotes “intrinsic”, T is the temperature in Kelvin, ΔE is the band gap energy, and
kB=1.38×10-23 J/K is the Boltzmann’s constant. The T3/2 term is a phase space factor; from the
Uncertainty principle, the density of momentum states per unit volume dN /dV µ (2 p /
relativistic electrons/holes in a semiconductor:
(
)
) 3, so for non-
3
2 2mE
dN
m3 / 2 E 3 / 2
(5)
µ
~
3
3
dV
where m is the effective mass of the electrons/holes (which, because of interactions with the lattice, can
be very different from the rest mass of the electron). For a full derivation of equation (4), see Kittel’s
“Introduction to Solid State Physics”, Chapter 8.
At high temperatures, the temperature dependence of the conductivity is largely determined by
the exponential term in equation (4), since the mobilities in equation (1) vary only as a power law of the
temperature [see Adler (1964)], i.e. =CTB, where B and C are constants. So we expect the conductivity
to be of the form
(6)
Hall Effect
Consider the sample of p-type semiconductor with current density Jx flowing in the x-direction.
In the presence of a magnetic field B0 along the z-direction, the holes will experience a force (the
Lorentz force) driving them towards the bottom of the sample as shown in Figure 1.
Figure 1: The Hall effect for positive charge carriers.
This net charge displacement results in the bottom surface becoming positively charged, and the top
surface becoming negatively charged. This process obviously cannot continue indefinitely since it would
generate an infinite field in the y-direction. What happens is that the electric field Ey adjusts itself so that
the force on the holes due to this field exactly cancels the Lorentz force:
eE y = ev x Bz
(7)
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where vx is the hole velocity in m/s, and Bz is the magnetic field in Tesla. Now the current density in the
x-direction is:
Jx = epv x
(8)
so that
Ey =
J x Bz
ep
(9)
we define the Hall coefficient as:
RH =
Ey
Jx Bz
=
1
ep
(10)
for p-type semiconductors. In a similar manner it can be shown that for an n-type semiconductor, in
which the charge carriers are electrons with charge -e, the Hall coefficient is
1
1
=(11)
-en
en
Note that the Hall coefficient has opposite signs for n and p-type semiconductors. The carrier
concentration p or n can be determined from equations (10) or (11), and when combined in equation (2)
or (3) with the measured conductivity, the carrier mobility h or e can be determined.
RH =
In any real semiconductor, both holes and electrons contribute to the conductivity. If one type of
carrier is dominant, it is called the majority carrier and the other type is referred to as the minority
carrier. The contribution to the conductivity from minority carriers cannot always be neglected, and
may need to be included in your analysis.
Safety Reminders
 Do not touch any part of the sample or sample card when the current is turned on; be aware that the
sample may remain hot for some time after the heater is turned off.
 Do not touch the electromagnet connectors when the electromagnetic power supply is on.
 Keep metallic or electronic objects away from the magnets. It is not always obvious which objects
might be forcefully attracted, e.g. all current Canadian coins are ferromagnetic. Also keep in mind
that strong magnetic fields may erase magnetic strips on credit, bank or other similar cards, or
damage electronic devices.
NOTE: This is not a complete list of all hazards; we cannot warn against every possible dangerous
stupidity, e.g. opening plugged-in electrical equipment, juggling cryostats, …. Experimenters must
constantly use common sense to assess and avoid risks, e.g. if you spill liquid on the floor it will become
slippery, sharp edges may cut you, …. If you are unsure whether something is safe, ask the supervising
professor, the lab technologist, or the lab coordinator. If an accident or incident happens, you must let
us know. More safety information is available at http://www.ehs.utoronto.ca/resources.htm.
Experiment
Measurements are made using the van der Pauw method described at
http://www.nist.gov/pml/div683/hall_resistivity.cfm, using a square semiconductor sample with leads
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attached at each corner. By flowing current through two leads, and measuring the voltage across the
other two, at different temperatures and in different magnetic fields, the resistivity (ρ), Hall coefficient
(RH), Hall mobility (h and e.), majority carrier sign (i.e. electrons or holes), carrier density (n or p),
and band-gap of the sample can be determined.
Hall Coefficient
Following the NIST procedures described starting at
http://www.nist.gov/pml/div683/hall_resistivity.cfm#hall, measure the Hall Coefficient, RH, of both nand p-doped samples at room temperature. Do not exceed the maximum current recommended by the
Advanced Physics Lab technologists. One can use either a permanent magnet or an electromagnet for
these measurements. It may also be interesting to measure the Hall coefficient of an undoped sample.
Measure the magnetic field with a gaussmeter and determine the carrier densities.
Conductivity Versus Temperature
Following the NIST procedures described starting at
http://www.nist.gov/pml/div683/hall_resistivity.cfm#resistivity, determine the variation of the
conductivity with temperature over the range from room temperature to > 100 C. The interesting parts
of the conductivity versus temperature curve occur below about 120°C. Overheating the sample may
damage the Hall Probe!
•
•
•
•
•
Indicate the ranges of temperature over which the samples exhibit the characteristics of intrinsic
and extrinsic conductivity.
Is equation (4) consistent with the data over any temperature region?
Is equation (6) consistent with the data?
Determine the energy gap for both the n- and p- doped samples. Are the gaps all the same? It
may be interesting to also determine the gap for an undoped sample.
Can you determine any other interesting parameter? e.g. the mobility temperature exponent B, or
the mean electron/hole effective mass (memp)1/2.
 Conductivity depends on both mobility and carrier density, while the Hall coefficient
depends just on the carrier density, so measuring both the Hall coefficient and the
conductivity as a function of temperature may allow you to determine the temperature
dependence of the mobility and carrier density separately.
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Bibliography
R. B. Adler et al., Introduction to Semiconductor Physics, Wiley, 1964 (QC 612 S4A4).
C. Kittel, Introduction to Solid State Physics, Wiley, 6th Edition, 1986 (QC 171 K5).
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